Question

The first & second stage of the two stages rocket separately weigh $100kg$ and $10kg$ and contain $800kg$ and $90kg$ fuel respectively. If the exhaust velocity of gases is $2km/sec$ then velocity of rocket after second stage is (nearly) $({\log }_{10}5=0.6990)$ (neglect gravity)

• A. $7.8\times {10}^{3}m/s$
• B. $9.3\times {10}^{3}m/s$
• C. $2.4\times {10}^{3}m/s$
• D. $6.1\times {10}^{3}m/s$

Answer 1

The correct option is A $7.8\times {10}^{3}m/s$

The initial velocity of the rocket, $v_0=0$

The rocket velocity after the exhaustion of the first stage is, $v=uln\frac{m_0}{m}$

Here,

$m_0=100+10+800+90=1000kg$

and

$m=100+10+90=200kg$

Thus,

$|v|=2ln5$

$=2\times 1.609=3.281km/s$

The second stage becomes operative when the first is detached from it.

Thus,

$u_0=3.218km/s$

$m_0=10+90=100kg$

$m=10kg$

Thus

$|v|=3.22+2ln\frac{100}{10}$

$=3.22+2\times \left(2.3\right)=7.82km/s=7.8\times {10}^{3}m/s$